Gear Trains

When the driver and follower must rotate in the same direction, an idler gear is inserted between them. The movement ratio for this arrangement is:

i_M = \frac{n_D}{n_F} = \frac{t_I}{t_D} \cdot \frac{t_F}{t_I} = \frac{t_F}{t_D}

where:

$t_I$ = number of teeth on the idler gear.

This shows the idler gear does not change the overall gear ratio ( $t_F / t_D$ ), but it reverses the rotation direction back to the original.

Bicycle Gearing

A bicycle's drivetrain is a practical application of gear trains. The distance traveled by the wheel per pedal revolution is calculated using:

l = \pi d \frac{n_D}{n_F} = \pi d \frac{t_D}{t_F}

where:

$l$ = distance traveled by the wheel
$d$ = outer diameter of the wheel
$t_D$ = number of teeth on the pedal sprocket (chainring)
$t_F$ = number of teeth on the wheel sprocket (cog)

Example Calculation

For a bike with a 26-inch wheel diameter, a 24-tooth front chainring, and a 34-tooth rear cog:

Wheel circumference $c = \pi d \approx 81.7$ in.
Distance per pedal stroke $l = (81.7 \text{ in}) \times (24/34) \approx 57.7$ inches.

For the highest gear (42T front, 14T rear):

Distance per pedal stroke $l = (81.7 \text{ in}) \times (42/14) \approx 244.8$ inches.

References

Original Source